Abstract by Jalen Morgan
J.D. Mireles James
Parameterization method for unstable manifolds of nonlinear waves
In previous projects we've worked on, we've identified both stable and unstable wave solutions for various PDEs. The stable waves are solutions that converge back to a constant shape when perturbed. The unstable waves, however, lose their shape when even the smallest error is introduced. In this project we've focused on the unstable waves and how they behave when given a perturbation. Specifically, we look at the eigenfunction of the waves to evolve them in the direction of greatest instability. We do this using a parameterization method we designed for the problem. To confirm the accuracy of our time evolution method, we compare with finite difference approximations.